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NavierStokes equationsFrom Wikipedia, the free encyclopedia

In physics, the NavierStokes equations/nvje stoks/, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of

viscous fluid substances. These balance equations arise from applying Newton's second law to fluid motion, together with the assumption that the stress

in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure termhence describing viscous flow. The

main difference between them and the simpler Euler equations for inviscid flow is that NavierStokes equations also in the Froude limit (no external

field) are not conservation equations, but rather a dissipative system, in the sense that they cannot be put into the quasilinear homogeneous form:

NavierStokes equations are useful because they describe the physics of many things of scientific and engineering interest. They may be used to model

the weather, ocean currents, water flow in a pipe and air flow around a wing. The NavierStokes equations in their full and simplified forms help with

the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of pollution, and many other things. Coupled with

Maxwell's equations they can be used to model and study magnetohydrodynamics.

The NavierStokes equations are also of great interest in a purely mathematical sense. Somewhat surprisingly, given their wide range of practical uses,

it has not yet been proven that in three dimensions solutions always exist (existence), or that if they do exist, then they do not contain any singularity

(they are smooth). These are called the NavierStokes existence and smoothness problems. The Clay Mathematics Institute has called this one of the

seven most important open problems in mathematics and has offered a US$1,000,000 prize for a solution or a counter-example. [1]

Contents

1 Flow velocity

2 General continuum equations

2.1 Convective acceleration

3 Incompressible flow

3.1 Discrete velocity

3.2 Pressure recovery

4 Compressible flow

5 Other equations

5.1 Continuity equation

6 Stream function for 2D equations

7 Properties

7.1 Nonlinearity

7.2 Turbulence

7.3 Applicability

8 Application to specific problems

9 Exact solutions of the NavierStokes equations

9.1 A three-dimensional steady-state vortex solution

10 Wyld diagrams

11 Representations

12 NavierStokes equations use in games

13 See also

14 Notes

15 References

16 External links
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An example of convection. Though

the flow may be steady (time-

independent), the fluid decelerates as

it moves down the diverging duct

(assuming incompressible or subsonic

compressible flow), hence there is an

acceleration happening over position.

Flow velocity

The solution of the NavierStokes equations is a flow velocity. It is a field, since it is defined at every point in a region of space and an interval of time.

Once the velocity field is calculated, other quantities of interest, such as pressure or temperature, may be found. This is different from what one

normally sees in classical mechanics, where solutions are typically trajectories of position of a particle or deflection of a continuum. Studying velocity

instead of position makes more sense for a fluid however for visualization purposes one can compute various trajectories.

General continuum equations

The NavierStokes momentum equation can be derived as a particular form of the Cauchy momentum equation. In an inertial frame of reference, the

conservation form of the equations of continuum motion is:[2]

Cauchy momentum equation(conservation form)

here

is the density,is the flow velocity,is the del operator.

is the pressureis the identity matrixis the deviatoric stress tensor, which has order two,represents body accelerations (per unit mass) acting on the continuum, for example gravity, inertial accelerations, electric field acceleration,

and so on.

The left side of the equation describes acceleration, and may be composed of time-dependent, convective, and hydrostatic effects (also the effects of

non-inertial coordinates if present). The right side of the equation is in effect a summation of body forces (such as gravity) and divergence of deviatoric

stress.

In the Eulerian forms it is apparent that the assumption of no deviatoric stress brings Cauchy equations to the Euler equations. All non-relativistic

balance equations, such as the NavierStokes equations, can be derived by beginning with the Cauchy equations and specifying the stress tensor

through a constitutive relation. By expressing the shear tensor in terms of viscosity and fluid velocity, and assuming constant density and viscosity, the

Cauchy equations will lead to the NavierStokes equations.

The incompressible case is simpler than the compressible one so for didactical purpose it should be presented before. However, the compressible caseis the most general framework of NavierStokes equations so where not specified, NavierStokes equations are intended to be compressibleNavier

Stokes equations.

Convective acceleration

A significant feature of Cauchy equation and consequently all other continuum equations (including Euler and

NavierStokes) is the presence of convective acceleration: the effect of time-independent acceleration of a flow

ith respect to space. While individual fluid particles indeed experience time-dependent acceleration, the

convective acceleration of the flow field is a spatial effect, one example being fluid speeding up in a nozzle.

Incompressible flow

The incompressible momentum NavierStokes equation result from the following assumptions on the Cauchy

stress tensor:[3]

the stress is Galileian invariant: it does not depend directly on the flow velocity, but only on spatialderivatives of the flow velocity. So the stress variable is the tensor gradient .

the fluid is assumed to be isotropic, as with gases and simple liquids, and consequently is an isotropictensor furthermore, since the deviatoric stress tensor can be expressed in terms of the dynamic viscosity

:

Stokes's stress constitutive equation(expression used for incompressible elastic solids)

where Iis the identity tensor, and

is the rate-of-strain tensor. So this decomposition can be explicited as:[3]
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Stokes's stress constitutive equation(expression used for incompressible viscous fluids)

Dynamic viscosityneed not be constant in incompressible flows it can depend on density and on pressure. Any equation expliciting one of these

transport coefficient in the conservative variables is called an equation of state.[4]

The divergence of the deviatoric stress is given by:

Incompressibility rules out density and pressure waves like sound or shock waves, so this simplification is not useful if these phenomena are of interest.

The incompressible flow assumption typically holds well with all fluids at low Mach numbers (say up to about Mach 0.3), such as for modelling air

inds at normal temperatures.[5]For incompressible (uniform density 0) flows the following identity holds:

here wis the specific (with the sense ofper unit mass) thermodynamic work, the internal source term. Then the incompressible NavierStokesequations are best visualised by dividing for the density:

Incompressible NavierStokes equations(convective form)

Velocity profile (laminar flow)

, ,

for the - direction, simplify Navier-Stokes equation

integrate twice to find velocity profile with boundary conditions: , ,

From this equation, sub in your two boundary conditions to get 2 equations

Add and solve for B

Substitute and solve for A

Finally you get the velocity profile

in tensor notation:

Incompressible NavierStokes equations(convective form)
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here:

=

0is the kinematic viscosity

It is well worth observing the meaning of each term (compare to the Cauchy momentum equation):

The higher-order term, namely the shear stress divergence , has simply reduced to the vector laplacian term2u.[6]This laplacian term can beinterpreted as the difference between the velocity at a point and the mean velocity in a small surrounding volume. This implies that for a Newtonian

fluid viscosity operates as a diffusion of momentum, in much the same way as the heat conduction. In fact neglecting the convection term,

incompressible NavierStokes equations lead to a vector diffusion equation (namely Stokes equations), but in general the convection term is present, so

incompressible NavierStokes equations belong to the class of convection-diffusion equations.

In the usual case of an external field being a conservative field:

by defining the hydraulic head:

one can finally condense the whole source in one term, arriving to the incompressible Navier-Stokes equation with conservative external field:

The incompressible NavierStokes equations with conservative external field is the fundamental equation of hydraulics. The domain for these

equations is commonly a 3 or less euclidean space, for which an orthogonal coordinate reference frame is usually set to explicit the system of scalar

partial derivative equations to be solved. In 3D orthogonal coordinate systems are 3: Cartesian, cylindrical, and spherical. Expressing the Navier-Stokes

vector equation in Cartesian coordinates is quite straightforward and not much influenced by the number of dimensions of the euclidean space

employed, and this is the case also for the first-order terms (like the variation and convection ones) also in non-cartesian orthogonal coordinate

systems. But for the higher order terms (the two coming from the divergence of the deviatoric stress that distinguish NavierStokes equations from

Euler equations) some tensor calculus is required for deducing an expression in non-cartesian orthogonal coordinate systems.

The incompressible NavierStokes equation is composite, the sum of two orthogonal equations,

here Sand Iare solenoidal and irrotational projection operators satisfying S+ I= 1and and fIare the non-conservative and conservativeparts of the body force. This result follows from the Helmholtz Theorem (also known as the fundamental theorem of vector calculus). The first

equation is a pressureless governing equation for the velocity, while the second equation for the pressure is a functional of the velocity and is related to

the pressure Poisson equation.

The explicit functional form of the projection operator in 3D is found from the Helmholtz Theorem:

ith a similar structure in 2D. Thus the governing equation is an integro-differential equation similar to Coulomb and Biot-Savart law, not convenient

for numerical computation.

An equivalent weak or variational form of the equation, proved to produce the same velocity solution as the NavierStokes equation,[7]is given by,

for divergence-free test functions wsatisfying appropriate boundary conditions. Here, the projections are accomplished by the orthogonality of thesolenoidal and irrotational function spaces. The discrete form of this is imminently suited to finite element computation of divergence-free flow, as we

shall see in the next section. There we will be able to address the question, "How does one specify pressure-driven (Poiseuille) problems with a

pressureless governing equation?"

The absence of pressure forces from the governing velocity equation demonstrates that the equation is not a dynamic one, but rather a kinematic

equation where the divergence-free condition serves the role of a conservation equation. This all would seem to refute the frequent statements that the

incompressible pressure enforces the divergence-free condition.

Discrete velocity
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With partitioning of the problem domain and defining basis functions on the partitioned domain, the discrete form of the governing equation is,

It is desirable to choose basis functions which reflect the essential feature of incompressible flow the elements must be divergence-free. While the

velocity is the variable of interest, the existence of the stream function or vector potential is necessary by the Helmholtz Theorem. Further, to

determine fluid flow in the absence of a pressure gradient, one can specify the difference of stream function values across a 2D channel, or the line

integral of the tangential component of the vector potential around the channel in 3D, the flow being given by Stokes' Theorem. Discussion will be

restricted to 2D in the following.

We further restrict discussion to continuous Hermite finite elements which have at least first-derivative degrees-of-freedom. With this, one can draw alarge number of candidate triangular and rectangular elements from the plate-bending literature. These elements have derivatives as components of the

gradient. In 2D, the gradient and curl of a scalar are clearly orthogonal, given by the expressions,

Adopting continuous plate-bending elements, interchanging the derivative degrees-of-freedom and changing the sign of the appropriate one gives many

families of stream function elements.

Taking the curl of the scalar stream function elements gives divergence-free velocity elements.[8][9]The requirement that the stream function elements

be continuous assures that the normal component of the velocity is continuous across element interfaces, all that is necessary for vanishing divergence

on these interfaces.

Boundary conditions are simple to apply. The stream function is constant on no-flow surfaces, with no-slip velocity conditions on surfaces. Stream

function differences across open channels determine the flow. No boundary conditions are necessary on open boundaries, though consistent values may

be used with some problems. These are all Dirichlet conditions.

The algebraic equations to be solved are simple to set up, but of course are non-linear, requiring iteration of the linearized equations.

Similar considerations apply to three-dimensions, but extension from 2D is not immediate because of the vector nature of the potential, and there exists

no simple relation between the gradient and the curl as was the case in 2D.

Pressure recovery

Recovering pressure from the velocity field is easy. The discrete weak equation for the pressure gradient is,

here the test/weight functions are irrotational. Any conforming scalar finite element may be used. However, the pressure gradient field may also be of

interest. In this case one can use scalar Hermite elements for the pressure. For the test/weight functions gione would choose the irrotational vector

elements obtained from the gradient of the pressure element.

Compressible flow

The NavierStokes equations result from the following assumptions on the stress tensor:[3]

the stress is Galileian invariant: it does not depend directly on the flow velocity, but only on spatial derivatives of the flow velocity. So the

stress variable is the tensor gradient u.

the stress is linearin this variable: (u) = C: (u),where Cis the fourth-order tensor representing the constant of proportionality, calledthe viscosity or elasticity tensor, and : is the double-dot product.

the fluid is assumed to be isotropic, as with gases and simple liquids, and consequently is an isotropic tensor furthermore, since the stress

tensor is symmetric, by Helmholtz decomposition it can be expressed in terms of two scalar Lam parameters, the bulk viscosity and thedynamic viscosity,as it is usual in linear elasticity:

Linear stress constitutive equation(expression used for elastic solid)

where Iis the identity tensor, (u) 12u+

12

(u)Tis the rate-of-strain tensor anduis the rate of expansion of the flow. So this

decomposition can be explicited as:

Since the trace of the rate-of-strain tensor in three dimensions is:

The trace of the stress tensor in three dimensions becomes:
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So by alternatively decomposing the stress tensor is into isotropicand deviatoricparts, as usual in fluid dynamics:[10]

Introducing the second viscosity ,

e arrive to the linear constitutive equation in the form usually employed in thermal hydraulics:[3]

Linear stress constitutive equation(expression used for fluids)

Both bulk viscosityand dynamic viscosityneed not be constant in general, they depend on density, on each other (the viscosity is expressed inpressure), and in compressible flows also on temperature. Any equation expliciting one of these transport coefficient in the conservation variables is

called an equation of state.[4]

By computing the divergence of the stress tensor, since the divergence of tensor uis2uand the divergence of tensor (u)Tis(u)., one

finally arrives to the compressible (most general) Navier-Stokes momentum equation:[11]

Navier-Stokes momentum equation(convective form)

Bulk viscosity is assumed to be constant, otherwise it should not be taken out of the last derivative. The effect of the volume viscosity is that the

mechanical pressure is not equivalent to the thermodynamic pressure:[12]

This difference is usually neglected, sometimes by explicitly assuming = 0, but it could have an impact in sound absorption and attenuation and shock

aves , see .[13]

For the special case of an incompressible flow, the pressure constrains the flow so that the volume of fluid elements is constant: isochoric flow

resulting in a solenoidal velocity field with u= 0.[14]

Other equations

The NavierStokes equations are strictly a statement of the balance of momentum. To fully describe fluid flow, more information is needed, how much

depending on the assumptions made. This additional information may include boundary data (no-slip, capillary surface, etc.), conservation of mass,

balance of energy, and/or an equation of state.

Continuity equation

Regardless of the flow assumptions, a statement of the conservation of mass is generally necessary. This is achieved through the mass continuity

equation, given in its most general form as:

or, using the substantive derivative:

It can be derived from using the Navier-Stokes Equation above.

In the example below we can assume to have a Newtonian Fluid as well as having and both be constant

Recall that mass continuity is simply the summation of the rate of mass in and the rate of mass out.

[Rate of Mass Accumulated] = [Rate of Mass In] - [Rate of Mass Out]
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Since there is no change in Pressure (P) over time, we have:

Recall that is a constant thus proving the divergence theorem above.

Stream function for 2D equations

Taking the curl of the NavierStokes equation results in the elimination of pressure. This is especially easy to see if 2D Cartesian flow is assumed (like

in the degenerate 3D case with and no dependence of anything onz), where the equations reduce to:

Differentiating the first with respect toy, the second with respect toxand subtracting the resulting equations will eliminate pressure and any

conservative force. Defining the stream function through

results in mass continuity being unconditionally satisfied (given the stream function is continuous), and then incompressible Newtonian 2D momentum

and mass conservation condense into one equation:

here is the (2D) biharmonic operator and is the kinematic viscosity, . We can also express this compactly using the Jacobian

determinant:

This single equation together with appropriate boundary conditions describes 2D fluid flow, taking only kinematic viscosity as a parameter. Note thatthe equation for creeping flow results when the left side is assumed zero.

In axisymmetric flow another stream function formulation, called the Stokes stream function, can be used to describe the velocity components of an

incompressible flow with one scalar function.

The incompressible NavierStokes equation is a differential algebraic equation, having the inconvenient feature that there is no explicit mechanism for

advancing the pressure in time. Consequently, much effort has been expended to eliminate the pressure from all or part of the computational process.

The stream function formulation eliminates the pressure but only in two dimensions and at the expense of introducing higher derivatives and

elimination of the velocity, which is the primary variable of interest.

Properties

Nonlinearity

The NavierStokes equations are nonlinear partial differential equations in the general case and so remain in almost every real situation. [15][16]In some

cases, such as one-dimensional flow and Stokes flow (or creeping flow), the equations can be simplified to linear equations. The nonlinearity makes

most problems difficult or impossible to solve and is the main contributor to the turbulence that the equations model.
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The nonlinearity is due to convective acceleration, which is an acceleration associated with the change in velocity over position. Hence, any convective

flow, whether turbulent or not, will involve nonlinearity. An example of convective but laminar (nonturbulent) flow would be the passage of a viscous

fluid (for example, oil) through a small converging nozzle. Such flows, whether exactly solvable or not, can often be thoroughly studied and

understood.[17]

Turbulence

Turbulence is the time-dependent chaotic behavior seen in many fluid flows. It is generally believed that it is due to the inertia of the fluid as a whole:

the culmination of time-dependent and convective acceleration hence flows where inertial effects are small tend to be laminar (the Reynolds number

quantifies how much the flow is affected by inertia). It is believed, though not known with certainty, that the NavierStokes equations describe

turbulence properly.

[18]

The numerical solution of the NavierStokes equations for turbulent flow is extremely difficult, and due to the significantly different mixing-length

scales that are involved in turbulent flow, the stable solution of this requires such a fine mesh resolution that the computational time becomes

significantly infeasible for calculation or direct numerical simulation. Attempts to solve turbulent flow using a laminar solver typically result in a time-

unsteady solution, which fails to converge appropriately. To counter this, time-averaged equations such as the Reynolds-averaged NavierStokes

equations (RANS), supplemented with turbulence models, are used in practical computational fluid dynamics (CFD) applications when modeling

turbulent flows. Some models include the Spalart-Allmaras, k- (k-omega), k- (k-epsilon), and SST models, which add a variety of additional

equations to bring closure to the RANS equations. Large eddy simulation (LES) can also be used to solve these equations numerically. This approach is

computationally more expensivein time and in computer memorythan RANS, but produces better results because it explicitly resolves the larger

turbulent scales.

Applicability

Together with supplemental equations (for example, conservation of mass) and well formulated boundary conditions, the NavierStokes equations

seem to model fluid motion accurately even turbulent flows seem (on average) to agree with real world observations.

The NavierStokes equations assume that the fluid being studied is a continuum (it is infinitely divisible and not composed of particles such as atoms

or molecules), and is not moving at relativistic velocities. At very small scales or under extreme conditions, real fluids made out of discrete molecules

ill produce results different from the continuous fluids modeled by the NavierStokes equations. Depending on the Knudsen number of the problem,

the Boltzmann equation may be a suitable replacement failing that, one may find the techniques of statistical mechanics sufficient or have to resort to

molecular dynamics.

Another limitation is simply the complicated nature of the equations. Time-tested formulations exist for common fluid families, but the application of

the NavierStokes equations to less common families tends to result in very complicated formulations and often to open research problems. For this

reason, these equations are usually rewritten for Newtonian fluids where the viscosity model is linear truly general models for the flow of other kinds

of fluids (such as blood) do not, as of 2012, exist .

Application to specific problems

The NavierStokes equations, even when written explicitly for specific fluids, are rather generic in nature and their proper application to specific

problems can be very diverse. This is partly because there is an enormous variety of problems that may be modeled, ranging from as simple as the

distribution of static pressure to as complicated as multiphase flow driven by surface tension.

Generally, application to specific problems begins with some flow assumptions and initial/boundary condition formulation, this may be followed by

scale analysis to further simplify the problem.

a) Assume steady, parallel, one dimensional, non-convective pressure-driven flow between parallel plates, the resulting scaled (dimensionless)

boundary value problem is:

The boundary condition is the no slip condition. This problem is easily solved for the flow field:

From this point onward more quantities of interest can be easily obtained, such as viscous drag force or net flow rate.

b) Difficulties may arise when the problem becomes slightly more complicated. A seemingly modest twist on the parallel flow above would be the

radial flow between parallel plates this involves convection and thus non-linearity. The velocity field may be represented by a function that must

satisfy:

This ordinary differential equation is what is obtained when the NavierStokes equations are written and the flow assumptions applied (additionally,the pressure gradient is solved for). The nonlinear term makes this a very difficult problem to solve analytically (a lengthy implicit solution may be

found which involves elliptic integrals and roots of cubic polynomials). Issues with the actual existence of solutions arise for R > 1.41 (approximately

this is not the square root of 2), the parameter R being the Reynolds number with appropriately chosen scales. [19]This is an example of flow

assumptions losing their applicability, and an example of the difficulty in "high" Reynolds number flows.[19]
http://-/?-http://-/?-https://en.wikipedia.org/wiki/Square_root_of_2https://en.wikipedia.org/wiki/Cubic_formulahttps://en.wikipedia.org/wiki/Elliptic_integralshttps://en.wikipedia.org/wiki/Implicit_functionhttps://en.wikipedia.org/wiki/Nonlinearhttps://en.wikipedia.org/wiki/Ordinary_differential_equationhttps://en.wikipedia.org/wiki/No_slip_conditionhttps://en.wikipedia.org/wiki/Boundary_value_problemhttps://en.wikipedia.org/wiki/Scale_analysis_(mathematics)https://en.wikipedia.org/wiki/Surface_tensionhttps://en.wikipedia.org/wiki/Multiphase_flowhttps://en.wikipedia.org/wiki/Linearhttps://en.wikipedia.org/wiki/Newtonian_fluidhttps://en.wikipedia.org/wiki/Molecular_dynamicshttps://en.wikipedia.org/wiki/Statistical_mechanicshttps://en.wikipedia.org/wiki/Boltzmann_equationhttps://en.wikipedia.org/wiki/Knudsen_numberhttps://en.wikipedia.org/wiki/Special_relativity#Relativistic_mechanicshttps://en.wikipedia.org/wiki/Continuum_mechanicshttps://en.wikipedia.org/wiki/Large_eddy_simulationhttps://en.wikipedia.org/wiki/Turbulence_kinetic_energyhttps://en.wikipedia.org/wiki/K-omega_turbulence_modelhttps://en.wikipedia.org/wiki/Spalart%E2%80%93Allmaras_turbulence_modelhttps://en.wikipedia.org/wiki/Computational_fluid_dynamicshttps://en.wikipedia.org/wiki/Reynolds-averaged_Navier%E2%80%93Stokes_equationshttps://en.wikipedia.org/wiki/Direct_numerical_simulationhttp://-/?-https://en.wikipedia.org/wiki/Reynolds_numberhttps://en.wikipedia.org/wiki/Inertiahttps://en.wikipedia.org/wiki/Chaos_theoryhttps://en.wikipedia.org/wiki/Turbulencehttp://-/?-https://en.wikipedia.org/wiki/Nozzlehttps://en.wikipedia.org/wiki/Laminar_flowhttps://en.wikipedia.org/wiki/Convective


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Visualization of a) parallel flow and b) radial flow.

Exact solutions of the NavierStokes equations

Some exact solutions to the NavierStokes equations exist. Examples of degenerate cases with the non-linear terms in the NavierStokes equations

equal to zero are Poiseuille flow, Couette flow and the oscillatory Stokes boundary layer. But also more interesting examples, solutions to the full

non-linear equations, exist for example the TaylorGreen vortex.[20][21][22]Note that the existence of these exact solutions does not imply they are

stable: turbulence may develop at higher Reynolds numbers.

Under additional assuptions, the component parts can be separated (http://www.claudino.webs.com/Navier%20Stokes%20Equations.pps).

For example, in the case of an unbounded planar domain with two-dimensional incompressible and stationary flow in polar

coordinates the velocity components and pressurepare:[23]

where A and B are arbitrary constants. This solution is valid in the domain r 1 and for .

In Cartesian coordinates, when the viscosity is zero ( ), this is:

For example, in the case of an unbounded Euclidean domain with three-dimensional

inco mpressible, stationary and with zero viscosity ( ) radial flow in

Cartesian coordinates the velocity vector and pressurepare:

There is a singularity at .

A three-dimensional steady-state vortex solution

A nice steady-state example with no singularities comes from considering the flow along the lines of a Hopf fibration. Let r be a constant radius to the

inner coil. One set of solutions is given by:[24]

A two dimensional example

A three-dimensional example
http://-/?-https://en.wikipedia.org/wiki/Hopf_fibrationhttps://en.wikipedia.org/wiki/Cartesian_coordinateshttp://-/?-https://en.wikipedia.org/wiki/Polar_coordinateshttp://www.claudino.webs.com/Navier%20Stokes%20Equations.ppshttp://-/?-http://-/?-http://-/?-https://en.wikipedia.org/wiki/Taylor%E2%80%93Green_vortexhttps://en.wikipedia.org/wiki/Stokes_boundary_layerhttps://en.wikipedia.org/wiki/Couette_flowhttps://en.wikipedia.org/wiki/Hagen-Poiseuille_equationhttps://en.wikipedia.org/wiki/File:NSConvection_vectorial.svg


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Some of the flow lines along a Hopf

fibration.

for arbitrary constants A and B. This is a solution in a non-viscous gas (compressible fluid) whose density,

velocities and pressure goes to zero far from the origin. (Note this is not a solution to the Clay Millennium

problem because that refers to incompressible fluids where is a constant, neither does it deal with the

uniqueness of the NavierStokes equations with respect to any turbulence properties.) It is also worth

pointing out that the components of the velocity vector are exactly those from the Pythagorean quadruple

parametrization. Other choices of density and pressure are possible with the same velocity field:

Another choice of pressure and density with the same velocity vector above is one where the pressure and density fall to zero at the origin

and are highest in the central loop at :

In fact in general there are simple solutions for any polynomial function f where the density is:

Wyld diagrams

Wyld diagramsare bookkeeping graphs that correspond to the NavierStokes equations via a perturbation expansion of the fundamental continuum

mechanics. Similar to the Feynman diagrams in quantum field theory, these diagrams are an extension of Keldysh's technique for nonequilibrium

processes in fluid dynamics. In other words, these diagrams assign graphs to the (often) turbulent phenomena in turbulent fluids by allowing correlated

and interacting fluid particles to obey stochastic processes associated to pseudo-random functions in probability distributions. [25]

Representations

From the general form of the Navier-Stokes, with the velocity vector expanded as , sometimes respectively named

, we may write the vector equation explicitly,

Note that gravity has been accounted for as a body force, and the values ofgx,gy,gzwill depend on the orientation of gravity with respect

to the chosen set of coordinates.

The continuity equation reads:

When the flow is incompressible, does not change for any fluid particle, and its material derivative vanishes: . The continuity

equation is reduced to:

Other choices of density and pressure

3D Car tesian coordinates
https://en.wikipedia.org/wiki/Material_derivativehttp://-/?-https://en.wikipedia.org/wiki/Probability_distributionhttps://en.wikipedia.org/wiki/Function_(mathematics)https://en.wikipedia.org/wiki/Pseudo-randomhttps://en.wikipedia.org/wiki/Stochastic_processeshttps://en.wikipedia.org/wiki/Correlation_functionhttps://en.wikipedia.org/wiki/Turbulencehttps://en.wikipedia.org/wiki/Graph_theoryhttps://en.wikipedia.org/wiki/Mstislav_Keldyshhttps://en.wikipedia.org/wiki/Quantum_field_theoryhttps://en.wikipedia.org/wiki/Feynman_diagramhttps://en.wikipedia.org/wiki/Continuum_mechanicshttps://en.wikipedia.org/wiki/Perturbation_theoryhttps://en.wikipedia.org/wiki/Graph_(mathematics)https://en.wikipedia.org/wiki/Pythagorean_quadruplehttps://en.wikipedia.org/wiki/Turbulencehttps://en.wikipedia.org/wiki/Hopf_fibrationhttps://en.wikipedia.org/wiki/File:Hopfkeyrings.jpg


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Thus, for the incompressible version of the Navier-Stokes equation the second part of the viscous terms fall away (see Incompressible

flow).

This system of four equations comprises the most commonly used and studied form. Though comparatively more compact than other

representations, this is still a nonlinear system of partial differential equations for which solutions are difficult to obtain.

A change of variables on the Cartesian equations will yield[5]the following momentum equations for r, , andz:

The gravity components will generally not be constants, however for most applications either the coordinates are chosen so that the gravity

components are constant or else it is assumed that gravity is counteracted by a pressure field (for example, flow in horizontal pipe is treated

normally without gravity and without a vertical pressure gradient). The con tinuity equation is:

This cylindrical representation of the incompressible NavierStokes equations is the second most commonly seen (the first being Cartesian

above). Cylindrical coordinates are chosen to take advantage of symmetry, so that a velocity component can disappear. A very commoncase is axisymmetric flow with the assumption of no tangential velocity ( ), and the remaining quantities are independent of :

In spherical coordinates, the r, , and momentum equations are[5](note the convention used: is polar angle, or colatitude,[26]0 ):

3D Cylindrical coordinates

3D Spherical coordinates
http://-/?-https://en.wikipedia.org/wiki/Colatitudehttp://-/?-https://en.wikipedia.org/wiki/Spherical_coordinateshttp://-/?-https://en.wikipedia.org/wiki/Partial_differential_equationshttps://en.wikipedia.org/wiki/Nonlinearhttps://en.wikipedia.org/wiki/Incompressible_flow


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Mass continuity will read:

These equations could be (slightly) compacted by, for example, factoring from the viscous terms. However, doing so would

undesirably alter the structure of the Laplacian and other quantities.

NavierStokes equations use in games

The NavierStokes equations are used extensively in video games in order to model a wide variety of natural phenomena. Simulations of small-scale

gaseous fluids, such as fire and smoke, are often based on the seminal paper "Real-Time Fluid Dynamics for Games" [27]by Jos Stam, which elaborates

one of the methods proposed in Stam's earlier, more famous paper "Stable Fluids" [28]from 1999. Stam proposes stable fluid simulation using a Navier

Stokes solution method from 1968, coupled with an unconditionally stable semi-Lagrangian advection scheme, as first proposed in 1992.

More recent implementations based upon this work run on the GPU as opposed to the CPU and achieve a much higher degree of performance. [29][30]

Many improvements have been proposed to Stam's original work, which suffers inherently from high numerical dissipation in both velocity and mass.

An introduction to interactive fluid simulation can be found in the 2007 ACM SIGGRAPH course, Fluid Simulation for Computer Animation.[31]

See also

Adhmar Jean Claude Barr de Saint-VenantBogoliubovBornGreenKirkwoodYvon hierarchy of equationsBoltzmann equationCauchy momentum equationConvectiondiffusion equationChapman-Enskog theoryChurchillBernstein equationCoand effectComputational fluid dynamicsEuler equationsHagenPoiseuille flow from the NavierStokes equationsDerivation of the NavierStokes equations

Non-dimensionalization and scaling of the NavierStokes equationsPressure-correction methodStokes equations

Reynolds transport theoremVlasov equation

Notes

1. Millennium Prize Problems, Clay Mathematics Institute, retrieved 2014-01-14

2. Batchelor (1967) pp. 137 & 142.3. Batchelor (1967) pp. 142148.4. Batchelor (1967) p. 165.

5. See Acheson (1990).6. Batchelor (1967) pp. 21 & 147.7. Temam, Roger (2001),NavierStokes Equations, Theory and Numerical Analysis, AMS Chelsea, pp. 107112

8. Holdeman, J.T. (2010), "A Hermite finite element method for incompressible fluid flow",Int. J. Numer. Meth. F Luids64(4): 376408,Bibcode:2010IJNMF..64..376H, doi:10.1002/fld.2154

9. Holdeman, J.T. Kim, J.W. (2010), "Computation of incompressible thermal flows using Hermite finite elements", Comput. Methods Appl. Mech. Engrg.199

(4952): 32973304, Bibcode:2010CMAME.199.3297H, doi:10.1016/j.cma.2010.06.03610. Chorin, Mardsen p.3311. Batchelor (1967) pp. 147 & 154.

12. Landau & Lifshitz (1987) pp. 4445, 19613. White (2006) p. 67.14. Batchelor (1967) p. 75.

15. Fluid Mechanics (Schaum's Series), M. Potter, D.C. Wiggert, Schaum's Outlines, McGraw-Hill (USA), 2008, ISBN 978-0-07-148781-816. Vectors, Tensors , and the basic Equations of Fluid Mechanics, R. Aris, Dover Publications, 1989, ISBN(10) 0-486-66110-517. McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, ISBN 0-07-051400-3

18. Encyclopaedia of Physics (2nd Edition), R. G. Lerner, G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3

19. Shah, Tasneem Mohammad (1972). "Analysis of the multigrid method". Published by Dr. TM Shah republished by The Smithsonian/NASA Astrophysics Data

System. Retrieved 8 January 2013.20. Wang, C.Y. (1991), "Exact solutions of the steady-state NavierStokes equations", Annual Review of Fluid Mechanics23: 159177,

Bibcode:1991AnRFM..23..159W, doi:10.1146/annurev.fl.23.010191.001111

21. Landau & Lifshitz (1987) pp. 7588.

22. Ethier, C. R. Steinman, D.A. (1994), "Exact fully 3D NavierStokes solutions for benchmarking",International Journal for Numerical Methods in F luids19 (5):369375, Bibcode:1994IJNMF..19..369E, doi:10.1002/fld.1650190502

23. Ladyzhenskaya, O.A. (1969), The Mathematical Theory of viscous Incompressible Flow (2nd ed.), p. preface, xi24. Kamchatno, A. M (1982), Topological solitons in magnetohydrodynamics(PDF)

25. McComb, W.D. (2008), Renormalization methods: A guide for beginners, Oxford University Press, ISBN 0-19-923652-6 pp. 121128.

26. Eric W. Weisstein (2005-10-26), Spherical Coordinates, MathWorld, retrieved 2008-01-2227. Stam, Jos (2003), Real-Time Fluid Dynamics for Games(PDF)
http://www.dgp.toronto.edu/people/stam/reality/Research/pdf/GDC03.pdfhttps://en.wikipedia.org/wiki/MathWorldhttp://mathworld.wolfram.com/SphericalCoordinates.htmlhttps://en.wikipedia.org/wiki/Eric_W._Weissteinhttps://en.wikipedia.org/wiki/Special:BookSources/0-19-923652-6https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://www.jetp.ac.ru/cgi-bin/dn/e_055_01_0069.pdfhttps://dx.doi.org/10.1002%2Ffld.1650190502https://en.wikipedia.org/wiki/Digital_object_identifierhttp://adsabs.harvard.edu/abs/1994IJNMF..19..369Ehttps://en.wikipedia.org/wiki/Bibcodehttps://dx.doi.org/10.1146%2Fannurev.fl.23.010191.001111https://en.wikipedia.org/wiki/Digital_object_identifierhttp://adsabs.harvard.edu/abs/1991AnRFM..23..159Whttps://en.wikipedia.org/wiki/Bibcodehttp://adsabs.harvard.edu/abs/1989STIN...9123418Shttps://en.wikipedia.org/wiki/Special:BookSources/0070514003https://en.wikipedia.org/wiki/Special:BookSources/9780071487818https://dx.doi.org/10.1016%2Fj.cma.2010.06.036https://en.wikipedia.org/wiki/Digital_object_identifierhttp://adsabs.harvard.edu/abs/2010CMAME.199.3297Hhttps://en.wikipedia.org/wiki/Bibcodehttps://en.wikipedia.org/w/index.php?title=Jin-Whan_Kim&action=edit&redlink=1https://en.wikipedia.org/w/index.php?title=J.T._Holdeman&action=edit&redlink=1https://dx.doi.org/10.1002%2Ffld.2154https://en.wikipedia.org/wiki/Digital_object_identifierhttp://adsabs.harvard.edu/abs/2010IJNMF..64..376Hhttps://en.wikipedia.org/wiki/Bibcodehttps://en.wikipedia.org/w/index.php?title=J.T._Holdeman&action=edit&redlink=1https://en.wikipedia.org/wiki/Roger_Temamhttp://www.claymath.org/millennium-problemshttps://en.wikipedia.org/wiki/Vlasov_equationhttps://en.wikipedia.org/wiki/Reynolds_transport_theoremhttps://en.wikipedia.org/wiki/Stokes_flowhttps://en.wikipedia.org/wiki/Pressure-correction_methodhttps://en.wikipedia.org/wiki/Non-dimensionalization_and_scaling_of_the_Navier%E2%80%93Stokes_equationshttps://en.wikipedia.org/wiki/Derivation_of_the_Navier%E2%80%93Stokes_equationshttps://en.wikipedia.org/wiki/Hagen%E2%80%93Poiseuille_flow_from_the_Navier%E2%80%93Stokes_equationshttps://en.wikipedia.org/wiki/Euler_equations_(fluid_dynamics)https://en.wikipedia.org/wiki/Computational_fluid_dynamicshttps://en.wikipedia.org/wiki/Coand%C4%83_effecthttps://en.wikipedia.org/wiki/Churchill%E2%80%93Bernstein_equationhttps://en.wikipedia.org/wiki/Chapman-Enskog_theoryhttps://en.wikipedia.org/wiki/Convection%E2%80%93diffusion_equationhttps://en.wikipedia.org/wiki/Cauchy_momentum_equationhttps://en.wikipedia.org/wiki/Boltzmann_equationhttps://en.wikipedia.org/wiki/BBGKY_hierarchyhttps://en.wikipedia.org/wiki/Adh%C3%A9mar_Jean_Claude_Barr%C3%A9_de_Saint-Venanthttp://-/?-https://en.wikipedia.org/wiki/ACM_SIGGRAPHhttp://-/?-http://-/?-http://-/?-https://en.wikipedia.org/wiki/Jos_Stamhttp://-/?-


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28. Stam, Jos (1999), Stable Fluids(PDF)

29. Harris, Mark J. (2004), "38", GPUGems - Fast Fluid Dynamics Simulation on the GPU

30. Sander, P. Tatarchuck, N. Mitchell, J.L. (2007), "9.6", ShaderX5 - Explicit Early-Z Culling for Efficient Fluid Flow Simulation, pp. 553564

31. Robert Bridson Matthias Mller-Fischer. "Fluid Simulation for Computer Animation". www.cs.ubc.ca.

References

Acheson, D. J. (1990),Elementary Fluid Dynamics, Oxford Applied Mathematics and Computing Science Series, Oxford University Press,ISBN 0-19-859679-0Batchelor, G. K. (1967),An Introduction to Fluid Dynamics, Cambridge University Press, ISBN 0-521-66396-2Landau, L. D. Lifshitz, E. M. (1987),Fluid mechanics, Course of Theoretical Physics 6(2nd revised ed.), Pergamon Press, ISBN 0-08-033932-

8, OCLC 15017127Rhyming, Inge L. (1991),Dynamique des fluides, Presses polytechniques et universitaires romandesPolyanin, A. D. Kutepov, A. M. Vyazmin, A. V. Kazenin, D. A. (2002),Hydrodynamics, Mass and Heat Transfer in Chemical Engineering,Taylor & Francis, London, ISBN 0-415-27237-8Currie, I. G. (1974),Fundamental Mechanics of Fluids, McGraw-Hill, ISBN 0-07-015000-1V. Girault and P.A. Raviart.Finite Element Methods for NavierStokes Equations: Theory and Algorithms.Springer Series in ComputationalMathematics. Springer-Verlag, 1986.White, F. M. (2006), Viscous Fluid Flow, McGraw-Hill, ISBN 0-07-124493-XV. Girault and P.A. Raviart.Finite Element Methods for NavierStokes Equations: Theory and Algorithms.Springer Series in ComputationalMathematics. Springer-Verlag, 1986.

External links

Simplified derivation of the NavierStokes equations (http://www.allstar.fiu.edu/aero/Flow2.htm)

Millennium Prize problem description (http://www.claymath.org/sites/default/files/navierstokes.pdf)CFD online software list (http://www.cfd-online.com/Wiki/Codes) A compilation of codes, including NavierStokes solversOnline fluid dynamics simulator (http://nerget.com/fluidSim/) Solves the NavierStokes equation numerically and visualizes it using JavascriptThree-dimensional unsteady form of the NavierStokes equations (https://www.grc.nasa.gov/www/k-12/airplane/nseqs.html) Glenn ResearchCenter, NASA

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Categories: Concepts in physics Equations of fluid dynamics Aerodynamics Partial differential equations

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